Physical review. B, Condensed matter

n -spin correlation functions for the two-dimensional Ising model

John L. Richardson, Myron Bander

Published Feb 1, 1978• 4 citations

Abstract

It is noted that expressions for the $T={T}_{c}$ $n$-spin correlation functions of a two-dimensional Ising system obtained by continuum methods do agree with exact results for restricted geometries. This leads to the conjecture that the continuum result is true for all $n$ and all geometries. Speculation regarding the behavior of these functions away from $T={T}_{c}$ are made.

DOI: 10.1103/physrevb.17.1464

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Spin-spin correlation functions for the two-dimensional Ising model: Exact theory in the scaling region

Tai Tsun Wu, Barry M. McCoy, Craig A. Tracy +1 more

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Theory of Toeplitz Determinants and the Spin Correlations of the Two-Dimensional Ising Model. III

Hung Cheng, Tai Tsun Wu

Physical ReviewDec 10, 196791 citations

Multispin correlation functions for Ising models

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Two-Dimensional Ising Model as an Exactly Solvable Relativistic Quantum Field Theory: Explicit Formulas for n -Point Functions

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Correlation Functions for the Two-Dimensional Ising Model

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Duality and even number spin-correlation functions in the two dimensional square lattice ising model

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arXiv:cond-mat/0703602Mar 23, 20071 citation

Physical review. B, Condensed matter

n -spin correlation functions for the two-dimensional Ising model

John L. Richardson, Myron Bander

Published Feb 1, 1978• 4 citations

Publisher

Abstract

It is noted that expressions for the

T={T}_{c}

n

-spin correlation functions of a two-dimensional Ising system obtained by continuum methods do agree with exact results for restricted geometries. This leads to the conjecture that the continuum result is true for all

n

and all geometries. Speculation regarding the behavior of these functions away from

T={T}_{c}

are made.

DOI: 10.1103/physrevb.17.1464

Related papers

Spin-spin correlation functions for the two-dimensional Ising model: Exact theory in the scaling region

Tai Tsun Wu, Barry M. McCoy, Craig A. Tracy +1 more

We compute exactly the spin-spin correlation functions $〈{\ensuremath{\sigma}}_{0,0}{\ensuremath{\sigma}}_{M,N}〉$ for the two-dimensional Ising model on a square lattice in zero magnetic field for $T>{T}_{c}$ and $T<{T}_{c}$ . We then analyze the correlation functions in the scaling limit $T\ensuremath{\rightarrow}{T}_{c},{M}^{2}+{N}^{2}\ensuremath{\rightarrow}\ensuremath{\infty}$ such that $(T\ensuremath{-}{T}_{c})$ is fixed. In this scaling limit $〈{\ensuremath{\sigma}}_{0,0}{\ensuremath{\sigma}}_{M,N}〉={R}^{\ensuremath{-}\frac{1}{4}}{F}_{\ifmmode\pm\else\textpm\fi{}}(t)+{R}^{\ensuremath{-}\frac{5}{4}}{F}_{1\ifmmode\pm\else\textpm\fi{}}(t)+o({R}^{\ensuremath{-}\frac{5}{4}})$, where $t$ is the scaling variable $\frac{R}{\ensuremath{\xi}}$ and ${F}_{\ifmmode\pm\else\textpm\fi{}}(t)$ and ${F}_{1\ifmmode\pm\else\textpm\fi{}}(t)$ are the scaling functions ($\ensuremath{\xi}$ is the correlation length). We derive exact expressions for these scaling functions, in terms of a Painlev\'e function of the third kind and analyze both the small- and large- $t$ behavior. A table of values for ${F}_{\ifmmode\pm\else\textpm\fi{}}(t)$ (good to ten significant digits) is also given. As an application we computer the coefficients ${C}_{0\ifmmode\pm\else\textpm\fi{}}$ and ${C}_{1\ifmmode\pm\else\textpm\fi{}}$ in the expansion ${k}_{B}T\ensuremath{\chi}(T)={C}_{0\ifmmode\pm\else\textpm\fi{}}{|1\ensuremath{-}\frac{{T}_{c}}{T}|}^{\ensuremath{-}\frac{7}{4}}+{C}_{1\ifmmode\pm\else\textpm\fi{}}{|1\ensuremath{-}\frac{{T}_{c}}{T}|}^{\ensuremath{-}\frac{3}{4}}+O(1)$ of the zero-field susceptibility $\ensuremath{\chi}(T)$ as $T\ensuremath{\rightarrow}{T}_{c}^{\ifmmode\pm\else\textpm\fi{}}$.We compute exactly the spin-spin correlation functions $〈{\ensuremath{\sigma}}_{0,0}{\ensuremath{\sigma}}_{M,N}〉$ for the two-dimensional Ising model on a square lattice in zero magnetic field for $T>{T}_{c}$ and $T<{T}_{c}$. We then analyze the correlation functions in the scaling limit $T\ensuremath{\rightarrow}{T}_{c},{M}^{2}+{N}^{2}\ensuremath{\rightarrow}\ensuremath{\infty}$ such that $(T\ensuremath{-}{T}_{c})$ is fixed. In this scaling limit $〈{\ensuremath{\sigma}}_{0,0}{\ensuremath{\sigma}}_{M,N}〉={R}^{\ensuremath{-}\frac{1}{4}}{F}_{\ifmmode\pm\else\textpm\fi{}}(t)+{R}^{\ensuremath{-}\frac{5}{4}}{F}_{1\ifmmode\pm\else\textpm\fi{}}(t)+o({R}^{\ensuremath{-}\frac{5}{4}})$, where $t$ is the scaling variable $\frac{R}{\ensuremath{\xi}}$ and ${F}_{\ifmmode\pm\else\textpm\fi{}}(t)$ and ${F}_{1\ifmmode\pm\else\textpm\fi{}}(t)$ are the scaling functions ($\ensuremath{\xi}$ is the correlation length). We derive exact expressions for these scaling functions, in terms of a Painlev\'e function of the third kind and analyze both the small- and large-$t$ behavior. A table of values for ${F}_{\ifmmode\pm\else\textpm\fi{}}(t)$ (good to ten significant digits) is also given. As an application we computer the coefficients ${C}_{0\ifmmode\pm\else\textpm\fi{}}$ and ${C}_{1\ifmmode\pm\else\textpm\fi{}}$ in the expansion ${k}_{B}T\ensuremath{\chi}(T)={C}_{0\ifmmode\pm\else\textpm\fi{}}{|1\ensuremath{-}\frac{{T}_{c}}{T}|}^{\ensuremath{-}\frac{7}{4}}+{C}_{1\ifmmode\pm\else\textpm\fi{}}{|1\ensuremath{-}\frac{{T}_{c}}{T}|}^{\ensuremath{-}\frac{3}{4}}+O(1)$ of the zero-field susceptibility $\ensuremath{\chi}(T)$ as $T\ensuremath{\rightarrow}{T}_{c}^{\ifmmode\pm\else\textpm\fi{}}$.

Physical review. B, Solid stateJan 1, 1976694 citations

Theory of Toeplitz Determinants and the Spin Correlations of the Two-Dimensional Ising Model. III

Hung Cheng, Tai Tsun Wu

We study the asymptotic behavior, for large separations, of the spin-spin correlation function $〈{\ensuremath{\sigma}}_{0,0}{\ensuremath{\sigma}}_{M,N}〉$ in the two-dimensional Ising model, where the two spins are not necessarily on the same row. Besides the limiting value for infinite separation, which is the square of the spontaneous magnetization, we evaluate the two leading terms in the asymptotic expression in each of the two cases $T<{T}_{c}$ and $T>{T}_{c}$ . It is found that the nearest singularity of the generating function for the correlation is quite simple in the case $T>{T}_{c}$ , but much more complicated for $T<{T}_{c}$ . In an Appendix, we also give exactly in a very simple form the correlation $〈{\ensuremath{\sigma}}_{0,0}{\ensuremath{\sigma}}_{N,N}〉$ for symmetrical Ising lattice at the critical temperature ${T}_{c}$ .We study the asymptotic behavior, for large separations, of the spin-spin correlation function $〈{\ensuremath{\sigma}}_{0,0}{\ensuremath{\sigma}}_{M,N}〉$ in the two-dimensional Ising model, where the two spins are not necessarily on the same row. Besides the limiting value for infinite separation, which is the square of the spontaneous magnetization, we evaluate the two leading terms in the asymptotic expression in each of the two cases $T<{T}_{c}$ and $T>{T}_{c}$. It is found that the nearest singularity of the generating function for the correlation is quite simple in the case $T>{T}_{c}$, but much more complicated for $T<{T}_{c}$. In an Appendix, we also give exactly in a very simple form the correlation $〈{\ensuremath{\sigma}}_{0,0}{\ensuremath{\sigma}}_{N,N}〉$ for symmetrical Ising lattice at the critical temperature ${T}_{c}$.

Physical ReviewDec 10, 196791 citations

Multispin correlation functions for Ising models

Helen Au-Yang

We calculate the multispin correlation functions away from ${T}_{c}$ of a two-dimensional Ising lattice along the diagonal. We also present the result for a four-spin correlation function in the scaling limit. These results show that with slight modifications, the operator reduction formulae of Kadanoff hold.We calculate the multispin correlation functions away from ${T}_{c}$ of a two-dimensional Ising lattice along the diagonal. We also present the result for a four-spin correlation function in the scaling limit. These results show that with slight modifications, the operator reduction formulae of Kadanoff hold.

Physical review. B, Solid stateMar 1, 19779 citations

Two-Dimensional Ising Model as an Exactly Solvable Relativistic Quantum Field Theory: Explicit Formulas for n -Point Functions

Barry M. McCoy, Craig A. Tracy, Tai Tsun Wu

For both $T>{T}_{c}$ and $T<{T}_{c}$ , we report exact expressions for the $n$ -spin correlation function of the two-dimensional Ising model suitable for studying large separations between spins. In particular, the scaling limit of these correlation functions can be shown to exist and yields $n$ -point Schwinger functions of some relativistic quantum field theories.For both $T>{T}_{c}$ and $T<{T}_{c}$, we report exact expressions for the $n$-spin correlation function of the two-dimensional Ising model suitable for studying large separations between spins. In particular, the scaling limit of these correlation functions can be shown to exist and yields $n$-point Schwinger functions of some relativistic quantum field theories.

Physical Review LettersApr 11, 1977140 citations

Correlation Functions for the Two-Dimensional Ising Model

Robert Hecht

The energy-density---energy-density and energy-density---spin correlation functions are evaluated for the two-dimensional square Ising model. Results are obtained at zero magnetic field and near the critical point, i.e., for $\ensuremath{\epsilon}=(\frac{4J}{\mathrm{kT}})|(T\ensuremath{-}\frac{{T}_{c}}{{T}_{c}})|\ensuremath{\ll}1.$ The correlation functions are ${〈{E}_{R}{E}_{0}〉\ensuremath{-}{〈E〉}^{2},={(\frac{2J}{\ensuremath{\pi}})}^{2}{\ensuremath{\epsilon}}^{2}[{{K}_{1}}^{2}(\ensuremath{\epsilon}R)\ensuremath{-}{{K}_{0}}^{2}(\ensuremath{\epsilon}R)],}{〈{E}_{R}{\ensuremath{\sigma}}_{0}〉\ensuremath{-}〈E〉〈\ensuremath{\sigma}〉,=(\frac{2J}{\ensuremath{\pi}})\ensuremath{\epsilon}〈\ensuremath{\sigma}〉\ensuremath{\int}{2\ensuremath{\epsilon}R}^{\ensuremath{\infty}}\mathrm{ds} {s}^{\ensuremath{-}2}{e}^{\ensuremath{-}8},}$ where $R$ is the distance between the points in question and ${K}_{0}$ and ${K}_{1}$ are modified Bessel functions of the second kind. Very similar results hold if the coupling constants are different in the two directions. The above results agree in form with the structure predicted by the "scaling-law" approach.The energy-density---energy-density and energy-density---spin correlation functions are evaluated for the two-dimensional square Ising model. Results are obtained at zero magnetic field and near the critical point, i.e., for $\ensuremath{\epsilon}=(\frac{4J}{\mathrm{kT}})|(T\ensuremath{-}\frac{{T}_{c}}{{T}_{c}})|\ensuremath{\ll}1.$ The correlation functions are ${〈{E}_{R}{E}_{0}〉\ensuremath{-}{〈E〉}^{2},={(\frac{2J}{\ensuremath{\pi}})}^{2}{\ensuremath{\epsilon}}^{2}[{{K}_{1}}^{2}(\ensuremath{\epsilon}R)\ensuremath{-}{{K}_{0}}^{2}(\ensuremath{\epsilon}R)],}{〈{E}_{R}{\ensuremath{\sigma}}_{0}〉\ensuremath{-}〈E〉〈\ensuremath{\sigma}〉,=(\frac{2J}{\ensuremath{\pi}})\ensuremath{\epsilon}〈\ensuremath{\sigma}〉\ensuremath{\int}{2\ensuremath{\epsilon}R}^{\ensuremath{\infty}}\mathrm{ds} {s}^{\ensuremath{-}2}{e}^{\ensuremath{-}8},}$ where $R$ is the distance between the points in question and ${K}_{0}$ and ${K}_{1}$ are modified Bessel functions of the second kind. Very similar results hold if the coupling constants are different in the two directions. The above results agree in form with the structure predicted by the "scaling-law" approach.

Physical ReviewJun 10, 196770 citations

Duality and even number spin-correlation functions in the two dimensional square lattice ising model

Ghosh, Ranjan Kumar

The Kramers-Wannier duality is shown to hold for all the even number spin correlation functions of the two dimensional square lattice Ising model in the sense that the high temperature $(T>T_{c})$ expressions for these correlation functions are transformed into the low temperature $(TThe Kramers-Wannier duality is shown to hold for all the even number spin correlation functions of the two dimensional square lattice Ising model in the sense that the high temperature $(T>T_{c})$ expressions for these correlation functions are transformed into the low temperature $(T

arXiv:cond-mat/0703602Mar 23, 20071 citation

It is noted that expressions for the

T={T}_{c}

n

n

and all geometries. Speculation regarding the behavior of these functions away from

T={T}_{c}

are made.

T>{T}_{c}

and

T<{T}_{c}

. We then analyze the correlation functions in the scaling limit $T\ensuremath{\rightarrow}{T}_{c},{M}^{2}+{N}^{2}\ensuremath{\rightarrow}\ensuremath{\infty}$ such that $(T\ensuremath{-}{T}_{c})$ is fixed. In this scaling limit $〈{\ensuremath{\sigma}}_{0,0}{\ensuremath{\sigma}}_{M,N}〉={R}^{\ensuremath{-}\frac{1}{4}}{F}_{\ifmmode\pm\else\textpm\fi{}}(t)+{R}^{\ensuremath{-}\frac{5}{4}}{F}_{1\ifmmode\pm\else\textpm\fi{}}(t)+o({R}^{\ensuremath{-}\frac{5}{4}})$, where

t

is the scaling variable $\frac{R}{\ensuremath{\xi}}$ and ${F}_{\ifmmode\pm\else\textpm\fi{}}(t)$ and ${F}_{1\ifmmode\pm\else\textpm\fi{}}(t)$ are the scaling functions ($\ensuremath{\xi}$ is the correlation length). We derive exact expressions for these scaling functions, in terms of a Painlev\'e function of the third kind and analyze both the small- and large-

t

behavior. A table of values for ${F}_{\ifmmode\pm\else\textpm\fi{}}(t)$ (good to ten significant digits) is also given. As an application we computer the coefficients ${C}_{0\ifmmode\pm\else\textpm\fi{}}$ and ${C}_{1\ifmmode\pm\else\textpm\fi{}}$ in the expansion ${k}_{B}T\ensuremath{\chi}(T)={C}_{0\ifmmode\pm\else\textpm\fi{}}{|1\ensuremath{-}\frac{{T}_{c}}{T}|}^{\ensuremath{-}\frac{7}{4}}+{C}_{1\ifmmode\pm\else\textpm\fi{}}{|1\ensuremath{-}\frac{{T}_{c}}{T}|}^{\ensuremath{-}\frac{3}{4}}+O(1)$ of the zero-field susceptibility $\ensuremath{\chi}(T)$ as $T\ensuremath{\rightarrow}{T}_{c}^{\ifmmode\pm\else\textpm\fi{}}$.

T<{T}_{c}

and

T>{T}_{c}

. It is found that the nearest singularity of the generating function for the correlation is quite simple in the case

T>{T}_{c}

, but much more complicated for

T<{T}_{c}

. In an Appendix, we also give exactly in a very simple form the correlation $〈{\ensuremath{\sigma}}_{0,0}{\ensuremath{\sigma}}_{N,N}〉$ for symmetrical Ising lattice at the critical temperature

{T}_{c}

We calculate the multispin correlation functions away from

{T}_{c}

of a two-dimensional Ising lattice along the diagonal. We also present the result for a four-spin correlation function in the scaling limit. These results show that with slight modifications, the operator reduction formulae of Kadanoff hold.

For both

T>{T}_{c}

and

T<{T}_{c}

, we report exact expressions for the

n

-spin correlation function of the two-dimensional Ising model suitable for studying large separations between spins. In particular, the scaling limit of these correlation functions can be shown to exist and yields

n

-point Schwinger functions of some relativistic quantum field theories.

R

is the distance between the points in question and

{K}_{0}

and

{K}_{1}

are modified Bessel functions of the second kind. Very similar results hold if the coupling constants are different in the two directions. The above results agree in form with the structure predicted by the "scaling-law" approach.

The Kramers-Wannier duality is shown to hold for all the even number spin correlation functions of the two dimensional square lattice Ising model in the sense that the high temperature

(T>T_{c})

expressions for these correlation functions are transformed into the low temperature $(T